Damon L. Bahr

This survey-based study examined how the intensity of the beliefs of novice inservice teachers about reform-based mathematics instruction changed as they crossed the threshold into their first year of teaching. It also investigated the relationship between the influence of new mentors, or transitional faculty, including mentor teachers, instructional coaches, and administrators, along with curriculum materials, and the changes belief intensity. Results obtained indicated belief changes were highly influenced by transitional faculty and curriculum materials. This study highlights the importance of creating systems to improve mentor selection and training so the work of university personnel is not undone during the novices first year of teaching.

The dispositions of preservice elementary education teachers toward reform-oriented mathematics education were surveyed before and after an extended pre-student teaching practicum. During the practicum, university and school-based personnel served a clinical faculty as they supported the preservice teachers practicum experiences. The preservice teachers perceptions of the clinical facultys dispositions were also surveyed. Relationships between changes in the dispositions of the preservice teachers and their perceptions of the clinical faculty were discovered thus highlighting the important influence clinical faculty wield as mathematics teacher educators.

This paper describes a study of observed relationships between the design of a preservice elementary mathematics methods course with accompanying field practicum and changes in the extent to which participating preservice teachers identified themselves with the mathematics reform movement after becoming inservice teachers. The curriculum of the course with its embedded field practicum experiences was designed to support preservice teachers in interweaving methods coursework and pedagogical instruction with classroom practice. University and elementary school structures were interwoven by conducting weekly mathematics methods course sessions at a school sites followed immediately by related practicum experiences in classrooms in the school. The interweaving of conceptualizations about mathematics teaching and learning was facilitated as the methods instructor provided professional development for classroom teachers simultaneously with the methods course, encouraging the teachers to create reform-friendly classroom environments. Survey data from 68 preservice teachers involved in the project were analysed to identify relationships between course/practicum experiences and the degree to which they as inservice teachers, 1-3 years following the course/practicum, identified with a reform mathematics perspective, as well as to examine factors influencing those relationships.

The purpose of this study was to compare changes in beliefs of two groups of preservice teachers involved in two types of opportunities to immediately apply methods for teaching accompanying an elementary mathematics methods course. Students in one group applied the methods learned in class through weekly 30-minute peer teaching sessions, while students in the other group worked for 45 minutes weekly with elementary students in public school classrooms where traditional pedagogy was normally practiced. The intensity of the beliefs about the nature of mathematics and of mathematical work held by these methods students was measured using the Integrating Mathematics and Pedagogy (IMAP) Web-Based Beliefs Survey (Philipp et al., 2007) as a pre- and post-assessment. While both groups saw significant change in belief intensity across measurement occasions favoring a reform perspective, a significantly greater change was experienced by the group who applied methods in classrooms, despite the traditional practice that usually occurred in them. The authors hypothesize this greater change resulted from the benefits associated with working with children and from the instructor support that may have tended to nullify the effects of teaching in a classroom where traditional pedagogy was the norm.

According to the equating guidelines, a set of common items should be a mini version of the total<br>test in terms of content and statistical representation (Kolen & Brennan, 2004). Differences<br>between vertical scaling and equating would suggest that these guidelines may not apply to<br>vertical scaling in the same way that they apply to equating. This study investigated how well the<br>guideline of content and construct representation was maintained while evaluating two stability<br>assessment criteria (Robust z and 0.3-logit difference). The results indicated that linking sets that<br>were not totally representative of the full test forms produced different vertical scales than the<br>linking sets that were most representative of the full test forms. The results also showed that<br>large disparities in the composition of linking sets produced statistically significant differences in<br>the growth patterns of the resulting vertical scales, but small disparities in the composition of<br>linking sets produced very similar vertical scales. Overall, the Robust z procedure was a more<br>conservative approach to flagging unstable items.

The NCTM Principles and Standards for School Mathematics calls for a balance between conceptual understanding and procedural knowledge. This study reports the results of a survey distributed to AMTE members in order to discover the opinions and practices of mathematics teacher educators regarding this balance. The authors conclude that there is wide disparity of views regarding the meaning of the terms "conceptual" and "procedural" as well as the meaning "balance" between the two, in terms of what constitutes mathematics, the learning and teaching of mathematics, and the assessment of mathematical proficiency.

As an integral part of a 2-year professional development program, 85 elementary teachers developed and administered grade-level specific mathematics performance assessments in number sense and operation sense?two assessments per grade level. Seven criteria based upon the NCTM Principles and Standards (2000) Learning Principle and Process Standards were created and then used to develop a rubric with a scoring hierarchy based upon the degree of assessor prompting required in order to ensure some degree of student success. Both assessments also included an instruction guide and suggested assessor questions or prompts to ensure that opportunities were provided for students to engage in performances that mirrored the criteria and addressed the components of number and operation sense as identified in the Principles and Standards. A ?Hierarchy of Numerical Complexity? to guide the determination of the students? level of numerical functioning relative to both tasks was then devised. A ?Place Value Hierarchy? based upon the Ross Five-Stage Place Value Understanding Model (1990, 1999) was developed with the number sense assessment, in order to designate varying levels of place value comprehension. An ?Algorithm Guide? based upon the Cognitively-Guided Instruction single digit and multi-digit invented algorithms (Carpenter, Fennema, Franke, Levi, & Empson, 1999) was also developed for labeling computational strategies in the operation sense assessment. Videotaped assessment administrations were utilized to teach assessment procedures and to enhance inter-rater reliability. Statistical analyses using Cronbach?s Alpha were used to establish that reliability among teachers.

Two groups of teachers who engaged in similar professional development were distinguished by the type of curriculum materials they used to guide their instructional practice. One group used a reform curricula and the other used a more traditional curricula. Using the Integrating Mathematics and Pedagogy Beliefs Survey, pre and post assessments of the intensity of the teachers? beliefs concerning mathematics reform were made. Analyses revealed that despite significant differences in belief intensity prior to professional development, there were no differences between the two groups on the post assessment. This suggests the relative strength of professional development in promoting reform compared to curriculum materials.

The CMI Framework was developed over several years of collaborative efforts between professors from four departments (Educational Leadership, Mathematics, Mathematics Education, and Teacher Education) at Brigham Young University and representatives from five surrounding school districts. The Framework was designed to provide access to reform-based pedagogical strategies for K-12 mathematics instructors, while building upon good strategies from traditional instruction. The Framework does not advocate a "balance" between these two approaches to instruction--although it does accommodate the goals of both--but rather a flow of experiences from individually constructed ideas towards a community of shared mathematical discourse. The Framework consists of three major components: a Teaching Cycle, a Learning Cycle, and a Continuum of Mathematical Understanding.

Effects of a practicum-based elementary mathematics methods course on the beliefs of preservice teachers regarding conceptual knowledge in school mathematics were explored using a pre-post design. The intensity of those beliefs was assessed before and after the methods course using the IMAP Web-Based Beliefs Survey, an instrument constructed by the ?Integrating Mathematics and Pedagogy? (IMAP) research group at San Diego State University. The IMAP Beliefs Survey assesses belief intensity by asking respondents to react to and analyze written and video cases (Ambrose, Clement, Philipp, & Chauvot, 2004; Philipp, Ambrose, Clement, Sowder, Schappelle, Sowder, Chauvot, & Thanheiser, 2005). Its use with preservice teachers enrolled in mathematics content courses for elementary education majors has been previously reported (Ambrose et al., 2004). In the current study, significant pre-post effects regarding belief changes were obtained in the desired direction, thus suggesting the appropriateness of conducting a future, more extensive experimental investigation of the effects of a practicum-based methods course on teacher beliefs.

A 2-year school-based mathematics professional development program is described and evaluated after its first year of implementation. Included in this program as its first course was a unique methods course in elementary education involving both preservice students and inservice teachers who cooperatively studied and applied reform pedagogy. The program resulted from the collaborative efforts of two institutions of higher education, a neighboring school district, the principal and teachers of one school within that district, and the state office of education. Evaluation of the first year of the program consisted of assessing the beliefs and perceptions of both preservice students and inservice teachers, along with an assessment of the mathematical achievement of the children within the classes of those teachers. Pre- and post-assessments of the preservice students and inservice teachers? beliefs regarding reform pedagogy were administered using the IMAP [Integrating Mathematics and Pedagogy] Web-Based Beliefs Survey (2001). Likert scale surveys were used to assess perceptions regarding course climate and participant relationships from both teacher groups. The mathematical achievement of children was assessed in three ways: The Wide Range Achievement Test-3 (Stone, Jastak, & Wilkinson, 1995), the Utah state criterion-referenced assessment, and performance assessments developed specifically for use at the school. Data obtained from all sources indicated positive effects upon teachers and children, thus providing substantial evidence in support of both the value of the methods course itself and the overall professional development program. An additional evaluation will be conducted following the second year of the program.

Teaching and learning are processes that are so interdependent, it is difficult to study one without considering the other. One purpose of the Comprehensive Mathematics Instruction Framework is to highlight this interconnection between teaching and learning to help teachers teach for deep mathematical understanding. In this paper we analyze a teaching episode using the CMI Framework as a tool for making sense of the work of the students and the teacher in the classroom.

The CMI Framework was developed over several years of collaborative efforts between professors from four departments (Educational Leadership, Mathematics, Mathematics Education, and Teacher Education) at Brigham Young University and representatives from five surrounding school districts. The Framework was designed to provide access to reform-based pedagogical strategies for K-12 mathematics instructors, while building upon good strategies from traditional instruction. The Framework does not advocate a "balance" between these two approaches to instruction--although it does accommodate the goals of both--but rather a flow of experiences from individually constructed ideas towards a community of shared mathematical discourse. The Framework consists of three major components: a Teaching Cycle, a Learning Cycle, and a Continuum of Mathematical Understanding.